MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__incr(X)) -> incr(X)
            head(cons(X,XS)) -> X
            incr(X) -> n__incr(X)
            incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
            nats() -> cons(0(),n__incr(nats()))
            odds() -> incr(pairs())
            pairs() -> cons(0(),n__incr(odds()))
            tail(cons(X,XS)) -> activate(XS)
        - Signature:
            {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1} / {0/0,cons/2,n__incr/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,head,incr,nats,odds,pairs
            ,tail} and constructors {0,cons,n__incr,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__incr(X)) -> c_2(incr#(X))
          head#(cons(X,XS)) -> c_3()
          incr#(X) -> c_4()
          incr#(cons(X,XS)) -> c_5(activate#(XS))
          nats#() -> c_6(nats#())
          odds#() -> c_7(incr#(pairs()),pairs#())
          pairs#() -> c_8(odds#())
          tail#(cons(X,XS)) -> c_9(activate#(XS))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__incr(X)) -> c_2(incr#(X))
            head#(cons(X,XS)) -> c_3()
            incr#(X) -> c_4()
            incr#(cons(X,XS)) -> c_5(activate#(XS))
            nats#() -> c_6(nats#())
            odds#() -> c_7(incr#(pairs()),pairs#())
            pairs#() -> c_8(odds#())
            tail#(cons(X,XS)) -> c_9(activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__incr(X)) -> incr(X)
            head(cons(X,XS)) -> X
            incr(X) -> n__incr(X)
            incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
            nats() -> cons(0(),n__incr(nats()))
            odds() -> incr(pairs())
            pairs() -> cons(0(),n__incr(odds()))
            tail(cons(X,XS)) -> activate(XS)
        - Signature:
            {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1,activate#/1,head#/1,incr#/1,nats#/0,odds#/0,pairs#/0
            ,tail#/1} / {0/0,cons/2,n__incr/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,head#,incr#,nats#,odds#,pairs#
            ,tail#} and constructors {0,cons,n__incr,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__incr(X)) -> incr(X)
          incr(X) -> n__incr(X)
          incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
          odds() -> incr(pairs())
          pairs() -> cons(0(),n__incr(odds()))
          activate#(X) -> c_1()
          activate#(n__incr(X)) -> c_2(incr#(X))
          head#(cons(X,XS)) -> c_3()
          incr#(X) -> c_4()
          incr#(cons(X,XS)) -> c_5(activate#(XS))
          nats#() -> c_6(nats#())
          odds#() -> c_7(incr#(pairs()),pairs#())
          pairs#() -> c_8(odds#())
          tail#(cons(X,XS)) -> c_9(activate#(XS))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__incr(X)) -> c_2(incr#(X))
            head#(cons(X,XS)) -> c_3()
            incr#(X) -> c_4()
            incr#(cons(X,XS)) -> c_5(activate#(XS))
            nats#() -> c_6(nats#())
            odds#() -> c_7(incr#(pairs()),pairs#())
            pairs#() -> c_8(odds#())
            tail#(cons(X,XS)) -> c_9(activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__incr(X)) -> incr(X)
            incr(X) -> n__incr(X)
            incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
            odds() -> incr(pairs())
            pairs() -> cons(0(),n__incr(odds()))
        - Signature:
            {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1,activate#/1,head#/1,incr#/1,nats#/0,odds#/0,pairs#/0
            ,tail#/1} / {0/0,cons/2,n__incr/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,head#,incr#,nats#,odds#,pairs#
            ,tail#} and constructors {0,cons,n__incr,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4}
        by application of
          Pre({1,3,4}) = {2,5,7,9}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__incr(X)) -> c_2(incr#(X))
          3: head#(cons(X,XS)) -> c_3()
          4: incr#(X) -> c_4()
          5: incr#(cons(X,XS)) -> c_5(activate#(XS))
          6: nats#() -> c_6(nats#())
          7: odds#() -> c_7(incr#(pairs()),pairs#())
          8: pairs#() -> c_8(odds#())
          9: tail#(cons(X,XS)) -> c_9(activate#(XS))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__incr(X)) -> c_2(incr#(X))
            incr#(cons(X,XS)) -> c_5(activate#(XS))
            nats#() -> c_6(nats#())
            odds#() -> c_7(incr#(pairs()),pairs#())
            pairs#() -> c_8(odds#())
            tail#(cons(X,XS)) -> c_9(activate#(XS))
        - Weak DPs:
            activate#(X) -> c_1()
            head#(cons(X,XS)) -> c_3()
            incr#(X) -> c_4()
        - Weak TRS:
            activate(X) -> X
            activate(n__incr(X)) -> incr(X)
            incr(X) -> n__incr(X)
            incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
            odds() -> incr(pairs())
            pairs() -> cons(0(),n__incr(odds()))
        - Signature:
            {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1,activate#/1,head#/1,incr#/1,nats#/0,odds#/0,pairs#/0
            ,tail#/1} / {0/0,cons/2,n__incr/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,head#,incr#,nats#,odds#,pairs#
            ,tail#} and constructors {0,cons,n__incr,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__incr(X)) -> c_2(incr#(X))
             -->_1 incr#(cons(X,XS)) -> c_5(activate#(XS)):2
             -->_1 incr#(X) -> c_4():9
          
          2:S:incr#(cons(X,XS)) -> c_5(activate#(XS))
             -->_1 activate#(X) -> c_1():7
             -->_1 activate#(n__incr(X)) -> c_2(incr#(X)):1
          
          3:S:nats#() -> c_6(nats#())
             -->_1 nats#() -> c_6(nats#()):3
          
          4:S:odds#() -> c_7(incr#(pairs()),pairs#())
             -->_2 pairs#() -> c_8(odds#()):5
             -->_1 incr#(X) -> c_4():9
             -->_1 incr#(cons(X,XS)) -> c_5(activate#(XS)):2
          
          5:S:pairs#() -> c_8(odds#())
             -->_1 odds#() -> c_7(incr#(pairs()),pairs#()):4
          
          6:S:tail#(cons(X,XS)) -> c_9(activate#(XS))
             -->_1 activate#(X) -> c_1():7
             -->_1 activate#(n__incr(X)) -> c_2(incr#(X)):1
          
          7:W:activate#(X) -> c_1()
             
          
          8:W:head#(cons(X,XS)) -> c_3()
             
          
          9:W:incr#(X) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: head#(cons(X,XS)) -> c_3()
          9: incr#(X) -> c_4()
          7: activate#(X) -> c_1()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__incr(X)) -> c_2(incr#(X))
            incr#(cons(X,XS)) -> c_5(activate#(XS))
            nats#() -> c_6(nats#())
            odds#() -> c_7(incr#(pairs()),pairs#())
            pairs#() -> c_8(odds#())
            tail#(cons(X,XS)) -> c_9(activate#(XS))
        - Weak TRS:
            activate(X) -> X
            activate(n__incr(X)) -> incr(X)
            incr(X) -> n__incr(X)
            incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
            odds() -> incr(pairs())
            pairs() -> cons(0(),n__incr(odds()))
        - Signature:
            {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1,activate#/1,head#/1,incr#/1,nats#/0,odds#/0,pairs#/0
            ,tail#/1} / {0/0,cons/2,n__incr/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,head#,incr#,nats#,odds#,pairs#
            ,tail#} and constructors {0,cons,n__incr,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:activate#(n__incr(X)) -> c_2(incr#(X))
           -->_1 incr#(cons(X,XS)) -> c_5(activate#(XS)):2
        
        2:S:incr#(cons(X,XS)) -> c_5(activate#(XS))
           -->_1 activate#(n__incr(X)) -> c_2(incr#(X)):1
        
        3:S:nats#() -> c_6(nats#())
           -->_1 nats#() -> c_6(nats#()):3
        
        4:S:odds#() -> c_7(incr#(pairs()),pairs#())
           -->_2 pairs#() -> c_8(odds#()):5
           -->_1 incr#(cons(X,XS)) -> c_5(activate#(XS)):2
        
        5:S:pairs#() -> c_8(odds#())
           -->_1 odds#() -> c_7(incr#(pairs()),pairs#()):4
        
        6:S:tail#(cons(X,XS)) -> c_9(activate#(XS))
           -->_1 activate#(n__incr(X)) -> c_2(incr#(X)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(6,tail#(cons(X,XS)) -> c_9(activate#(XS)))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          activate#(n__incr(X)) -> c_2(incr#(X))
          incr#(cons(X,XS)) -> c_5(activate#(XS))
          nats#() -> c_6(nats#())
          odds#() -> c_7(incr#(pairs()),pairs#())
          pairs#() -> c_8(odds#())
      - Weak TRS:
          activate(X) -> X
          activate(n__incr(X)) -> incr(X)
          incr(X) -> n__incr(X)
          incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
          odds() -> incr(pairs())
          pairs() -> cons(0(),n__incr(odds()))
      - Signature:
          {activate/1,head/1,incr/1,nats/0,odds/0,pairs/0,tail/1,activate#/1,head#/1,incr#/1,nats#/0,odds#/0,pairs#/0
          ,tail#/1} / {0/0,cons/2,n__incr/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/1,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {activate#,head#,incr#,nats#,odds#,pairs#
          ,tail#} and constructors {0,cons,n__incr,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE